Calculator
Formula Used
This calculator evaluates the joint likelihood of the sample under a chosen distribution. It also reports the log-likelihood, score, AIC, and BIC.
L(p) = ∏ p^xᵢ (1-p)^(1-xᵢ), with xᵢ ∈ {0,1}
L(p) = ∏ [C(n,xᵢ) p^xᵢ (1-p)^(n-xᵢ)], where n is trials per observation
L(λ) = ∏ [e^-λ λ^xᵢ / xᵢ!], with xᵢ = 0,1,2,...
L(λ) = ∏ [λ e^(-λxᵢ)], for xᵢ ≥ 0
L(μ,σ) = ∏ [(1 / (σ√(2π))) exp(-(xᵢ-μ)^2 / (2σ^2))], with σ > 0
The calculator plots a relative likelihood curve by subtracting the maximum log-likelihood before exponentiating. This keeps the graph numerically stable.
How to Use This Calculator
- Select the probability model that matches your data type.
- Paste observations into the sample field using commas or line breaks.
- Choose MLE mode to estimate parameters, or custom mode to test your own values.
- For the Binomial model, provide the number of trials per observation.
- Submit the form to see likelihood, log-likelihood, diagnostics, and the plot.
Example Data Table
| Distribution | Example Observations | Typical Parameter Input | Comment |
|---|---|---|---|
| Bernoulli | 1, 0, 1, 1, 0, 1 | p = 0.60 | Useful for binary outcomes like success or failure. |
| Binomial | 7, 6, 8, 5, 9 | trials = 10, p = 0.70 | Each observation counts successes from the same number of trials. |
| Poisson | 2, 3, 1, 0, 4, 2 | lambda = 2.00 | Good for event counts over equal exposure periods. |
| Exponential | 0.5, 1.2, 0.9, 1.8, 0.4 | lambda = 1.10 | Suitable for waiting times and time-between-events data. |
| Normal | 9.8, 10.4, 10.1, 9.9, 10.6 | mu = 10.00, sigma = 0.30 | Common for continuous measurements with symmetric spread. |
FAQs
1. What does the likelihood value mean?
It measures how plausible the observed sample is under the chosen parameter values. Larger likelihood means the data are better supported by that parameter setting.
2. Why is log-likelihood shown too?
Products of many small probabilities can underflow numerically. Log-likelihood avoids that problem and is easier to compare across parameter choices.
3. What is the difference between MLE and custom mode?
MLE estimates parameters directly from your sample. Custom mode lets you test any parameter values and compare them with the MLE benchmark.
4. Why can the raw likelihood become zero?
Very small products may round down to zero in floating-point arithmetic. That does not mean the model is impossible. Check the log-likelihood instead.
5. When should I use the Binomial model?
Use it when every observation records the number of successes out of a fixed number of trials, such as ten shots or twenty inspections.
6. What do AIC and BIC help with?
They summarize model fit while penalizing parameter count. Lower values generally indicate a more efficient balance between complexity and fit.
7. Why does the graph show relative likelihood?
Relative likelihood rescales the curve so the peak is one. That keeps the plot readable even when raw likelihood values are extremely tiny.
8. Can I use decimal observations for every distribution?
No. Bernoulli, Binomial, and Poisson require integer counts. Exponential and Normal accept continuous values, with Exponential also requiring nonnegative data.